The First Law for Boosted Kaluza-Klein Black Holes

TL;DR

The paper addresses the thermodynamics of stationary, non-rotating Kaluza-Klein black holes with momentum along the compact circle, deriving a first law where the conjugate to the compact length $\mathcal{L}$ is an effective tension $\hat{\mathcal{T}} = \mathcal{T} + v_H \mathcal{P}/\mathcal{L}$ and establishing two Smarr formulas. It uses a Hamiltonian perturbation framework to relate horizon and infinity data, and demonstrates that the effective tension remains well-behaved (positive for boosted strings) even when the ADM tension may become negative. The work provides explicit expressions for the ADM mass, tension, and momentum in terms of asymptotic data, derives the first law for both fixed and varying $\mathcal{L}$, and derives Smarr relations via scaling and Komar methods, followed by a Gibbs-Duhem-type relation for tension. Collectively, these results unify the thermodynamic structure across static and stationary KK black holes and clarify how momentum along the compact direction enters the first law and Smarr relations, with implications for black string stability and compactification physics. The analysis also highlights open questions, such as a complete Hamiltonian derivation of the tension Gibbs-Duhem relation in the stationary case.

Abstract

We study the thermodynamics of Kaluza-Klein black holes with momentum along the compact dimension, but vanishing angular momentum. These black holes are stationary, but non-rotating. We derive the first law for these spacetimes and find that the parameter conjugate to variations in the length of the compact direction is an effective tension, which generally differs from the ADM tension. For the boosted black string, this effective tension is always positive, while the ADM tension is negative for large boost parameter. We also derive two Smarr formulas, one that follows from time translation invariance, and a second one that holds only in the case of exact translation symmetry in the compact dimension. Finally, we show that the `tension first law' derived by Traschen and Fox in the static case has the form of a thermodynamic Gibbs-Duhem relation and give its extension in the stationary, non-rotating case.